Matter wave duality

Matter wave duality is one of the most fundamental theories which gives a direction to quantum physics from classical physics. It describes the dual nature of matter. That is, matter can behave like both matter and wave. This gives rise to the dual nature of the matter. The phenomenon of a beam of light that diffracted just like a wave is explained by this theory that every particle that is moving has a wavelength associated with it which explains the matter-wave nature or wave nature of the particle.

In 1924 a French physicist, Louis de Broglie, gave the proposal about the wave nature of the particle. It was observed that the electron, which we usually think of as a particle, may in some situations behave like a wave.

Observing the wave nature of the electron

De Broglie’s proposal of De Broglie’s theory was a bold one, made at a time when there was no direct experimental evidence of the matter-wave relation or the wave nature of the particle. But within a few years, his ideas were verified by diffraction experiments. There are also some other experiments like three-dimensional diffraction grating by X-ray, Bell Telephone Laboratories experiment and others like these which prove the wave nature of the microscopic particle which is in motion. 

Electron Waves And De Broglie’s Equation:

If a particle behaves like a wave, it should have a wavelength and a frequency. According to De Broglie’s theory-free particle with rest mass m, moving with non-relativistic speed v, should have a wavelength related to its momentum p=mv in exactly the same way as for a photon, as expressed by the equation:

                =h/p = h/mv

Where  = de Broglie wavelength of a particle

              h = Planck’s constant

              p = Particle’s momentum

              m = Particle’s mass

              v = Particle’s speed

If the particle’s speed is an appreciable fraction of the speed of light c, we replace mv in the equation with mv = m0v1-v2/c2, here m0 is the rest mass of the particle. The frequency f, according to de Broglie, is also related to the particle’s energy E in exactly the same way as for a photon:

      E=hf ,

 where, E = Energy of the particle

                h =  Planck’s constant

                 f = frequency

If the de Broglie picture is correct and the matter has wave-like aspects, you might wonder why we don’t see these aspects in everyday life. As an example, we know that waves diffract through a single slit. Yet when we walk through a doorway (A kind of a single slit), we don’t worry about our body diffracting!

The main reason we don’t see these effects on human scales is that Planck’s constant h has such a minuscule value. As a result, the de Broglie wavelengths of even the smallest ordinary objects that you can see are extremely small and the wave effects are unimportant. For example, what is the wavelength of a falling grain of sand? If the grain’s mass is 5×10-10 kg and its diameter is 0.07mm = 7×10-5 m, it will fall in the air with a terminal speed of about 0.4 m/s. The magnitude of its momentum is p = mv =(5×10-10)(0.4m/s)= 2×10-10kg.m/s. Now, if we calculate the wavelength, then it will be

= h/p

=6.626×10-34/2×10-10

= 3.308×10-24m    

So this wavelength is very small. That’s why we cannot observe this in real life. 

A more massive and high moving body has more momentum and even smaller de Broglie wavelength. The effect of such tiny wavelengths is so small that they are never noticed in our daily life.

Application of wave nature of the article

1. Electron microscope 

An electron microscope offers an important and interesting example of the dual nature of electrons. An electron beam can be used to form an image of an object much similar to that of light. So here, the wave nature of the electron comes into the picture for image formation, which is an application of De Broglie’s theory.

2. Atomic spectra 

Every neutral atom consists of at least one electron. So when a material is heated, it emits light and different materials have different types of light. This is because of the dual nature of the matter again, which is proved by De Broglie’s theory.

3. Bohr’s atomic model 

According to Bohr’s atomic model, the angular momentum of the electron is quantized.

Mathematically, mvr = nh.

Which is an imaginary proposal having no valid reason.

Heisenberg uncertainty principle:

So one thing must come to our mind here: what is the role of the Heisenberg uncertainty principle in matter-wave duality (wave nature of the matter). The Heisenberg uncertainty principle is one of the best principles that agree with the wave-particle duality as well as De Broglie’s equation. 

So let us understand the Heisenberg principle First,

“It states that measured value cannot assign to the position r and momentum p of a particle simultaneously with the unlimited precision. “

Mathematically, ∆x∆px  ≥ h/4π

or ∆x(m∆vx) ≥h/4π

or ∆x∆vx ≥h/4πm

Where, ∆x is the uncertainty in the position of the particle, 

             ∆p is the uncertainty in the momentum of the particle, 

             ∆v is the uncertainty in the velocity of the particle, 

              m is the mass of the particle, 

              h is Plank’s constant. 

So what is the relation of this with the matter-wave or wave nature of the particles? Let us understand it. 

One thing to be kept in mind is that both theories are not valid for the macroscopic world. In the Heisenberg Uncertainty principle, it states that momentum and position cannot be predicted at a time, which is based on the particle assumption. So, there should be a wave nature associated with this, which is discussed in quantum mechanics. 

Conclusion 

Matter waves-wave nature of matter is one of the reasons which violates the laws of classical mechanics. It gives the foundation to quantum mechanics. Every particle that is moving has a wavelength associated with it which explains the dual behaviour of the matter. Various applications of this equation are well established in the field and subjects of chemistry as well as physics.